Modelling, Simulation, and Control of a Flexible

Space Launch Vehicle

Muhammad Abdullah Aamer∗ , Qurat Ul Ain∗ , Ushbah Kaleem∗ , Hafiz Zeeshan Iqbal Khan∗† , and Jamshed Riaz∗
∗ Department of Aeronautics & Astronautics, Institute of Space Technology, Islamabad, Pakistan.
† Centers of Excellence in Science and Applied Technologies, Islamabad, Pakistan.
arXiv:2309.13032v1 [eess.SY] 22 Sep 2023

Abstract—Modern Space Launch Vehicles (SLVs), being slender system bandwidth. Proper determination of dominant vibration
in shape and due to the use of lightweight materials, are generally modes of launch vehicles is integral in designing the attitude
flexible in nature. This structural flexibility, when coupled with sensor controller [2]. Problems due to flexibility can be avoided by
and actuator dynamics, can adversely affect the control of SLV, which
may lead to vehicle instability and, in the worst-case scenario, to modifying the control system by including filters, stiffening
structural failure. This work focuses on modelling and simulation the sensor mounting structure or relocating sensors to locations
of rigid and flexible dynamics of an SLV and its interactions with where structural flexibility effects are minimum.
the control system. SpaceX’s Falcon 9 has been selected for this There are generally two approaches in literature for control
study. The flexible modes are calculated using modal analysis in design of a flexible SLV, i.e. gain stabilization and phase
Ansys. High-fidelity nonlinear simulation is developed which incor-
porates the flexible modes and their interactions with rigid degrees stabilization [3]. In phase stabilization, loop components and
of freedom. Moreover, linearized models are developed for flexible filters are selected such that the phase of structural feedback
body dynamics, over the complete trajectory until the first stage’s loop is 180◦ , whereas, in gain stabilization a filter that has
separation. Using classical control methods, attitude controllers, that deep notch is introduced at structural frequency, this keeps
keep the SLV on its desired trajectory, are developed, and multiple the loop gain well below unity. These filters such as notch
filters are designed to suppress the interactions of flexible dynamics.
The designed controllers along with filters are implemented in the filter, elliptic filter, etc. suppress vehicle’s flexibility and fuel
nonlinear simulation. Furthermore, to demonstrate the robustness of sloshing dynamics by providing gain attenuation and phase
designed controllers, Monte-Carlo simulations are carried out and stabilization [4].
results are presented. Notch filters requires prior knowledge of exact frequency of
Keywords—Space Launch Vehicles; Flexible Dynamics; Flexible flexible modes. Since the vibration frequency varies through-
Modes; Gain Stabilization; Notch Filters; Low Pass Filters; Elliptic
Filters out the trajectory as the propellant burns, it is difficult to
calculate the exact value of flexible mode frequencies [5]. This
makes the use of notch filters less practical. To overcome this
I. I NTRODUCTION
issue, other filters are suggested in literature, i.e. Elliptic filters
Curiosity of the mankind for space exploration has increased [6], Kalman filters [7], etc. Another solution is to use adaptive
the need for Space Launch Vehicles (SLVs). The minimum notch filters. Using sensor output signals, an adaptive notch
weight objectives of large but slender SLVs have led them to filter estimates exactly the frequency of the actual system.
exhibit structural flexibility. Structural flexibility depends upon The design parameters of the filter are updated continuously
vehicle fineness ratio, whose increase leads to issues related to match with the actual system parameters [8]. For vehicles
to vehicle dynamics and control. As flexibility increases, the that have two modes close to each other, the adaptive notch
modal frequencies of flexible modes get closer to the rigid filter can be extended to predict the frequency of these two
body modes that may result in rigid-elastic coupling. In addi- modes. This type of adaptive algorithm is useful for flexible
tion to displacement and acceleration due to rigid body motion, space launch vehicles that have low natural frequencies [5].
structural deflections can contribute to the net body motion. For advanced launch vehicle configurations with unstable
It is important to be able to model this change in behavior aerodynamics, high flexibility, liquid propellent sloshing, and
to avoid the deteriorating effects of its interference with the inertia effects of engine (tail wags) the classic control methods
flight control system. This control structure coupling causes along with filters are not effective in meeting robustness
the vehicle to deviate from desired performance resulting in margins. For such vehicles adaptive control techniques are
instability and in extreme scenarios, structural failure. usually employed [9], [10].
Flexibility affects the control loop in two ways. Firstly, In this paper, a nonlinear mathematical model for a flexible
it alters the output of the sensor to include the bending space launch vehicle is developed incorporating the effects of
frequencies in the feedback loop and secondly, it changes flexibility at sensor and actuator locations. SpaceX’s Falcon
the actuator position and actuator command angle resulting 9 is selected as reference SLV because most of relevant data
in altered control command to the vehicle [1]. In general, is available and remaining is obtained using CFD and Modal
this contribution of structural flexibility limits the control analysis. Based on the mathematical model, a high fidelity
nonlinear simulation is developed. Furthermore, this model is pitching moment coefficients, respectively, at each point in the
linearized around a trajectory to obtain a set of linear models. trajectory. Moreover, symmetry of the SLV shape is exploited
Using classical control theory, linear controller is designed to obtain the aerodynamic data for negative values of angle of
along with filters to mitigate the flexibility effects. In this work attack, and directional coefficients as a function of side-slip
we have designed both notch and elliptic filters and compared angle (β) and Mach number. Moreover, the rate derivatives e.g.
their performance and robustness. The designed controller Cmα̇ are assumed to be negligible and the dynamic derivatives
and filter are then implemented in nonlinear simulation, and (Cmq , Cnr , Clp ) are obtained from [12].
results are presented. Moreover, a Monte-Carlo analysis is Since the mass of space launch vehicles changes rapidly
preformed to compare the robustness of both filters towards as the propellent burns during the flight, the value of inertia
the uncertainties and variations in modal frequencies and mode and the location of center of gravity also change at each
shapes. point. Using CATIA the inertia matrix and the location of
The rest of the paper is organized as follows. Section II CG is determined for the CAD model. These parameters are
starts with aerodynamic and structural analysis results of Fal- calculated at the different fuel percentages and are tabulated
con 9, followed by nonlinear mathematical model of flexible in Table I. Interpolating these values gives us continuously
SLV along with its linearized version. Thereafter, Section changing inertial data corresponding to fuel percentage. This
III presents the designed controller and structural filters for is achieved by decreasing the mass of fuel tank in the CAD
gain stabilization of flexible modes. In Section IV nonlinear model as shown in Fig. 1.
simulation results are presented. Finally, the discussion is
B. Structural Analysis
concluded in Section V.
Before flexible body modeling, structural analysis needs to
II. M ATHEMATICAL M ODELLING be done to determine flexible modes through modal analysis.
To minimize unnecessary complexity, mathematical mod- Ansys Modal workbench is used to first generate a mesh
eling does not account for all of the variables present in the for a simplified Falcon 9 model and then modal analysis is
real system. To retain an accurate representation of the system, performed with both-ends-free boundary condition.
the trade-off is to make assumptions that incur minor errors Figure 2 shows an exaggerated view of the shape that the
in computations while considerably reducing complexity. In SLV takes under free vibrations when the first and second
this work, earth is assumed to be flat and non-rotating and mode bending frequency are excited individually. Our focus
considered as an inertial reference frame. Moreover, the fuel lies on the effects that accrue due to these first two bending
and oxidizer sloshing, and ‘tail wags dog’ effects are ignored modes. The frequencies for the two modes are tabulated in
while developing the dynamical model of the flexible SLV. Table II.
Before proceeding towards modelling, we introduce some To calculate the mode shapes, i.e. modal displacements and
notation which will be used in this work. I is inertial frame, their slopes, at the above-mentioned frequencies, a python
with origin at the launch point of vehicle. Br is rigid-body library pyAnsys [13] is used, which provides tools to import
fixed frame, centered at the c.g. of the SLV, with XB -axis and analyze Ansys output files. Damping ratio for each flexible
pointing towards nose and YB -axis pointing towards right. Bf mode is estimated using classical Rayleigh damping method
is the local flexible-body fixed frame at the sensor location. and using historical values from literature [3]. Thus, the
RFF12 represents the transformation matrix from F1 to F2 , Re (θ) displacement and slope of each node along the length of the
represents the rotation of angle θ about unit vector e, also e1 , SLV are computed for first two modes, in both y and z axis,
e2 , and e3 represents the unit vectors [1, 0, 0]⊤ , [0, 1, 0]⊤ , and and are shown in Fig. 3. It shows modal displacements (φ)
[0, 0, 1]⊤ , respectively. V represents velocity of c.g. of SLV in and slopes (σ) in y and z directions, along the x axis. Two
Br , and ω is angular velocity of Br w.r.t. I expressed in Br . locations along the x-axis are important, the location of nozzle
at 70 m from nose and the location of sensors at 15 m from
A. Aerodynamic and Inertial Data nose. These specific locations are denoted by subscripts T
As discussed earlier, SpaceX’s Falcon 9 is selected because and G, respectively, e.g. φYT represents value of φY at 70 m,
of availability of its material and engine propulsive data from and σZG represents value of σZ at 15 m, etc. This assumed
SpaceX website and other internet forums. Its slender body, sensor location is where deflection is minimum for first mode
with some parts made of composite structure, makes flexibility excitement.
a more prominent characteristic in its behavior compared to
metallic bodies of other SLVs. Since aerodynamic and inertial
data is not readily available on internet, we performed CFD
analysis of a CAD model of Falcon 9 from GrabCAD [11].
A simplified 2D version of this CAD model is used for CFD
analysis using Fluent. CFD is done at five values of angle
of attack (α) (0◦ , 2◦ , 4◦ , 6◦ , 8◦ ) and at five values of Mach
number (0.5, 1.5, 4, 7, 10). The data obtained is interpolated
to obtain the values of CL , CD , and Cm , i.e. lift, drag, Fig. 1: Exploded view of CAD model showing fuel tank
 TABLE I: Estimated inertial data at different fuel conditions

Parameter 100% fuel 75% fuel 50% fuel 25% fuel 0% fuel

mass (kg) 581726.686 511784.213 441841.74 371899.268 301956.795

CG [from nose] (mm) 38192.31 38939.239 38542.382 36340.671 31093.342
Jxx (kgm2 ) 1.516 × 106 1.408 × 106 1.299 × 106 1.191 × 106 1.083 × 106
Jyy , Jzz (kgm2 ) 2.545 × 108 2.516 × 108 2.506 × 108 2.388 × 108 1.94 × 108

(a) First bending mode (b) Second bending mode

Fig. 2: First two flexible modes

4.0 4.0
Disp. ϕy(x) [mm]

Disp. ϕz(x) [mm]

2.0 2.0
0.0 0.0
-2.0 -2.0
-4.0 Mode 1 Mode 3
-4.0
0.03 0.03
Slope σy(x) [deg]

Slope σz(x) [deg]

Mode 2 Mode 4
0.02 0.02
0.01 0.01
0.00 0.00
−0.01 −0.01
−0.02 −0.02
0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70
x - Location (from Nose) [m] x - Location (from Nose) [m]

Fig. 3: Variation of displacement and slope along the length of SLV

TABLE II: Frequency of 1st and 2nd Bending Mode where e3 = [0, 0, 1]⊤ , V = [u, v, w]⊤ represents body
velocity, ω = [p, q, r]⊤ represents body angular velocity, m
Mode Mode Type Frequency (Hz) Damping Ratio
and J are SLV’s mass and inertia matrix, respectively. Faero
7th 1st Bending Mode 4.293 0.0145 and Maero represents the aerodynamic forces and moments,
8th 1st Bending Mode 4.293 0.0145 respectively. Moreover, FT and τ represents the forces and
9th 2nd Bending Mode 11.559 0.0147 moments due to all engines and their gimbal deflections,
10th 2nd Bending Mode 11.559 0.0147 respectively.
First stage of the Falcon 9 SLV is powered by nine Merlin
engines [14], each of them is equipped with 2D gimballed
C. Rigid Body Dynamics nozzles. Consider the schematic shown in Fig. 4, we can write
total engine forces and moment as follows,
Considering the assumptions described earlier, and noting
that time variations of mass and inertia matrix are significant, 8
we can write the 6-DOF equation of motion of rigid dynamics X
FT = Fi (δi )
as follows,
i=0
8

−(L − xcg )
 (2)
m(V̇ + ω × V ) = mgRIBr e3 + Faero + FT
X
(1) τ=  −r sin λi  × Fi (δi )
˙ + J ω̇ + ω × Jω = Maero + τ
Jω i=0 r cos λi
 1) Flexibility Effects on Engine Gimbals: Structural flexi-
bility affects gimbal deflections. Forces and moments due to
all nine engines gets modified due the bending effects. For
that total engine forces and moments in Eq. (1) FT and τ will
get replaced by F̂T and τ̂ , respectively, which are expressed
as follows,
8
X
F̂T = F̂i (δi )
Engine
i=0
(a) Engines configuration (b) Gimbal angles 8

−(L − xcg )
  (7)
X
Fig. 4: Engines configuration and gimbal angles convention τ̂ =  −r sin λi  + φ⊤
Tξ
 × F̂i (δi )
i=0 r cos λi
where,
where, δ = [δ0⊤ , · · · , δ8⊤ ]⊤ , δi = [µi , ηi ]⊤ represents the ⊤
F̂i (δi ) = Re3 (σY⊤T ξ)Re2 (σZ ξ)Fi (δi ) (8)
gimbal deflections of ith engine, for all i ∈ [0, 8], L represents T

the length of SLV, and where, Fi (δi ) is the ith engine force vector without bending
Fi (δi ) = Re1 (λi )Re3 (µi )Re2 (ηi )Ti e1 , (3) as described in Eq. (3), and σYT and σZT are mode slopes at
engine gimbal location as defined in section II-B.
where Ti is the thrust of ith engine, and is assumed to be same 2) Flexibility Effects on Sensor Measurements: Another
for all engine, and it varies along the altitude as follows [14], key effect of flexible modes is their contribution in sensor
measurements, which if not properly taken care of, can get
Ti = 914.11 − 0.68P [kN], ∀ i ∈ [0, 8] (4)
fed back into the system through control and may results in
where is P represents the atmospheric pressure in kPa at a an unstable positive feedback loop. Let Bf be the local frame
given altitude. at sensor location, and aligned with the body frame Br in the
D. Flexible Modes Dynamics absence of bending. Then in case of bending we can write,
B B
In reality, a flexible structure contains an infinite degrees of RI f = RBrf RIBr (9)
freedom, making an exact analysis nearly impossible. How-
With small angle assumption, i.e. bending effects are small in
ever, by restricting the system to a finite number of degrees of B
magnitude, and using Rodrigues’ formula we can write RBrf
freedom, an approximate analysis can be accomplished. We
as follows
employed a finite element method (FEM) based strategy to
−σY⊤G ξ σZ ⊤
 
reduce the degrees of freedom. FEM model was developed in 1 G
ξ
B
Ansys Modal workbench as discussed in section II-B, only first RBrf ≈  σY⊤G ξ 1 0  (10)
⊤
four modes (first two bending modes) are considered [3], and −σZ G
ξ 0 1
the mass normalized mode shapes are obtained. Restricting Similarly, we can write body rates measured by the gyroscopes
to these selected modes, we can write flexible dynamics as ωm as follows,
follows,
⊤˙ B
⊤˙
8 ωm = σG ξ + RBrf ω ≈ σG ξ+ω (11)
ξ˙
    
0 I ξ X
= + φ Fi (δi ) (5)
ξ̈ −Ω2 −2ζΩ ξ˙ T where,
i=0
σYG ∈ Rnf ×3
 
σG = 0 σZG (12)
where ξ ∈ Rnf represents normalized deflection, Fi (δi )
as in Eq. (3), Ω = diag(Ω1 , Ω2 , · · · , Ωnf ), ζ = where σZG and σYG are the modal slopes at sensor location. It
diag(ζ1 , ζ2 , · · · , ζnf ). Where Ωj and ζj represents the modal is worth noting that since only bending modes are considered,
frequency and damping of jth flexible mode for all j in [1, nf ], roll angle and roll rate aren’t affected by flexibility.
here nf is the number of flexible modes considered. In this F. Control Allocation
paper we have selected nf = 4, and
Assuming same thrust for all engines (Ti = T ), and small
φT = 0 φYT φZT ∈ Rnf ×3
 
(6) gimbal deflections (δ) we can linearize Eq. (2) as,
 
where φYT and φZT represents the mode shapes as defined in δA
section II-B. τ ≅ T ΛGδ , T Λ δE  (13)
E. Interactions of Flexible and Rigid Modes δR
To develop the complete nonlinear model of flexible SLV, where,
we also need to model the effects of flexibility on rigid
 
8r 0 0
dynamics. We followed a similar approach as in [1], [15], Λ=0 −9(L − xcg ) 0 
and considered the following effects. 0 0 −9(L − xcg )
 Bode Diagram
From: deltaE (deg) To: theta (rad)
20
Rigid
0 Flexible

-20

-40

Magnitude (dB)
Fig. 5: Control architecture
-60

-80
 
1 −1 ∂τ
and G = [G0 , G1 · · · , G8 ], where Gi = Λ T , for all
∂δi
-100

i ∈ [0, 8]. So, using pseudo-inverse we can write the control -120

allocation as,  
-140

δA -160
10-3 10-2 10-1 100 101 102 103
δ = G† δE  (14) Frequency (rad/s)

δR Fig. 6: Comparison of rigid and flexible dynamics

−1
where G† = G⊤ GG⊤ .
G. Linearized Dynamics purpose, the design point for the controller is selected where
the system is most unstable. The rigid alone, and rigid and
In the process of control system design for the flexible space
flexible combined transfer functions are shown in Eqs. (17)
launch vehicle, it is necessary to obtain a set of linear model.
and (18), respectively. By incorporating four bending modes
In this regard, we linearized the complete flexible dynamics
at two distinct frequencies, the transfer function for the pitch
discussed in previous subsections, and assumed the decoupling
angle undergoes a significant transformation, as shown in Eq.
between different channels. The resulting short period approx-
(18). The resulting transfer function now includes eight addi-
imation of longitudinal model is same as presented in [1], as
tional poles, further complicating the dynamics of the system.
described below.
The inclusion of these bending modes in the transfer function
nf (r)
Zα Zδ X Zδ σ
T allows for a more accurate representation of the flexible effects
α̇ = α + q + E δE + ξr present in the space launch vehicle. By capturing the dynamics
V V r=1
V
nf (r)
! associated with bending modes at different frequencies, we
X (r) mZδ φT gain a more comprehensive understanding of the system’s
q̇ = Mα α + MδE δE + MδE σT + ξr (15)
r=1
Iy behavior and can design a controller that effectively addresses
these additional complexities.
θ̇ = q
(r) θ(s) −0.017725(s + 0.02853)
ξ¨r = −ωr2 ξr − 2ζωr ξ˙r + mZδE σT δE , ∀ r ∈ [1, nf ] = (17)
δE (s) s(s − 0.4067)(s + 0.4557)
Moreover, since the effects of flexibility on roll dynamics
are negligible, therefore, for roll channel, linear models are
−0.0142(s − 193.4)(s + 184.2)(s + 11.44)
same as that of rigid dynamics and are shown below.
θ(s) (s − 11.58)(s + 0.02848)
φ̇ = p = (18)
(16) δE (s) s(s + 0.4557)(s2 + 0.7826s + 727.6)
ṗ = Lp p + LδA δA (s − 0.4067)(s2 + 2.14s + 5275)
III. C ONTROL D ESIGN The comparison of the bode plots of transfer functions in
In this section linear control design is presented. The Eqs. (17) and (18) are shown in Fig. 6. Observing the Bode
linearized equations presented in section II-G, which account plots, distinct peaks can be observed in the red line for the
for the impact of flexibility effects, are used. These equations flexible body. These peaks correspond to the presence of two
capture the short period dynamics of the flexible space launch bending modes that are not adequately attenuated. The pres-
vehicle while considering the influence of its inherent flex- ence of these peaks signifies potential instability in the system,
ibility. Along a trajectory similar to that used in [16] a set as they introduce significant resonance and amplification at
of linear models are developed for different angle of attack specific frequencies associated with the bending modes.
conditions. Control architecture for pitch and yaw channels is To address the destabilizing peaks in the frequency response
shown in Fig. 5. Similar architecture but without any bending of the system, the incorporation of filters into the controller is
mode filters is used for roll channel. As pitch and yaw channels required. In the introduction section, various filters have been
are symmetric, so only pitch controller is presented. Moreover, mentioned. Among the available filter types, the elliptic filter
since roll dynamics and its control design is trivial so it is stands out as an ideal choice for our system due to its ability
skipped. to provide a sharp cut-off and effectively attenuate specific
The first step in a control design process is to select a frequencies associated with the bending modes [6]. Moreover,
single linear model, from of the set of linear models. For that elliptic filter has wide stopband allowing for effective rejection
 GM = 2.22 at 19.41 (rad/s) and PM = 63.44 at 2.63 (rad/s)
GM = 2.02 at 6.92 (rad/s) and PM = 40.93 at 2.50 (rad/s)
From: deltaE To: deltaE
100

Magnitude (dB)
50

-50

-100
270

180

Phase (deg)
90

0
Notch Filter
-90 Elliptic Filter
-180
10-3 10-2 10-1 100 101 102 103
Frequency (rad/s)

Fig. 7: Step response comparison: Notch and Elliptic filters Fig. 8: Frequency response comparison: Notch and Elliptic
filters

of frequencies outside the desired passband. On the other hand,

notch filters provide an efficient solution in terms of phase
lag if modal frequencies are accurately known. Therefore, in
this work we considered both elliptic and notch filters and
compared their performance. For notch filter we used a double
notch whose transfer function is shown below,
(s2 + 0.27s + 727.4)(s2 + 0.73s + 5275)
GN (s) = (19)
(s2 + 37.76s + 727.4)(s2 + 101.7s + 5275) Fig. 9: Nonlinear simulation
The elliptic filter parameters are selected as follows:
• First order (n) is 3. Falcon 9 is on a trajectory to the International Space Station,
• Passband Frequency (Wp) is 10 rad/s. thus the launch azimuth angle required from Cape Canaveral
• Passband ripple (Rp) is 1 dB. is 135◦ . Only trajectory, similar to that in [16], till first stage
• Stopband sttenuation (Rs) is 40 dB. separation is considered, that is about 165 seconds.
These values gives the following elliptic filter using MATLAB The nonlinear simulation results for both Notch and Elliptic
ellip command, filters, are shown in Figs. 11 and 12. In these simulations,
0.69201(s2 + 760.8) nominal values of all parameters are considered, and wind of
GE (s) = (20) 10 knots is applied along north and east direction. Fig. 11
(s + 5.237)(s2 + 4.545s + 100.5)
depicts the attitude angle and it can be seen that the SLV
Controller gains and compensator are tuned, and following follows the reference trajectory for pitch and yaw angle quite
values were selected, accurately, and the attitude errors remains within a fraction of
• KP = −114.5916 a degree over the complete trajectory. Similarly, as shown Fig.
0.1


• KP I (s) = −214.2862 1 + s 12 controller commands are also small. Moreover, both filters
This controller, along with both filters separately, was have same the performance in the nominal scenario.
analyzed on flexible models of the space launch vehicle. Fig. To compare the robustness of controller with each filter
7 shows comparison of step responses with both filters and towards the uncertainty in modal parameters, i.e. mode fre-
it can be seen that they are almost similar. However, form quencies and mode shapes, Monte-Carlo type simulations were
loop shape bode plot comparison shown in Fig. 8, we can see performed. Despite their similar nominal performance, with
that the notch filter provides better gain and phase margins as Elliptic filter closed loop remained stable upto ±34% variation
compared to the elliptic filter. in modal parameters, while with Notch filter it was stable only
upto ±3% variations. This observation, in contrast to fact that
IV. N ONLINEAR S IMULATION
controller with notch filters has more gain and phase margins,
The designed controller and filter are implemented in non- is consistent with the results in [6]. Which further emphasize
linear simulation developed in Simulink as shown in Fig. 9. that the elliptic filters are preferable for gain stabilization,
The dynamics block of the Falcon 9 SLV is shown in Fig. 10. while simple notch filters should be used only when modal
A reference trajectory for pitch and yaw angle is to be followed parameters are precisely known.
by Falcon 9. For pitch angle trajectory it is assumed that the
SLV remains completely vertical for the first 10 seconds and V. C ONCLUSION
then pitch angle decreases linearly from 90◦ to 40◦ for the rest This paper considered the problem of attitude control of
of the trajectory. Similarly for yaw angle it is assumed that a flexible SLV. Falcon 9 was selected for this study. A
 Fig. 10: Falcon 9 dynamics block

10 -4 Controller Commands
1

(deg)
0

A
-1
0 50 100 150

0.5
0
(deg)

-0.5
E

-1
-1.5
0 50 100 150

2
Notch Filter
(deg)

Elliptic Filter
1
R

0
0 50 100 150
Time (sec)

Fig. 11: Nominal performance: Euler angles and attitude errors Fig. 12: Nominal performance: Controller commands

nonlinear model of SLV dynamics (both rigid and flexible) and liquid propellant sloshing. Moreover, for off-nominal
was developed. Moreover, a classical PID type controller along conditions and in the presence of uncertainties adaptive control
with different filters was designed to mitigate the flexibility algorithms can be designed to suppress the destabilizing
effects during the ascent phase. Specifically, notch and elliptic effects.
filters were designed and compared. The simulation results
showed that the designed controller along with both filters R EFERENCES
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